In the simplest case, the word digital refers to the representation of
a quantity in numerical form and analog refers to a continuous physical
quantity. To digitize means to convert an analog physical
quantity into a numerical value. For example, if we represent the
intensity of a sound by numbers proportionally related to the intensity,
the analog
value of the intensity has been represented digitally. The accuracy of
the digital conversion depends upon the number of discrete numerical
values that can be assigned and the rate at which these numerical
measurements are made. For example, 4 numerical levels will represent
changes in the amplitude of
sound less accurately than 256 numerical levels and
a rate of 8 conversion/sec will be less accurate than
a rate of 10,000 conversions/sec.

The process for digitally coding sound by computer
was first developed in 1957 by Max Mathews
of Bell Telephone Laboratories in Murray Hill
(Mathews, 1963). Other advances in digital electronics
and microchips led to the development of the
first digital Pulse Code Modulation (PCM) audio
recorder in 1967 at the NHK Technical Research
Institute (Nakajima, 1983). This machine was a 12-
bit companded scheme (using a compression/expansion
of sound to improve dynamic range) with a 30
kHz sampling rate. Data were recorded on a one-
track, two-head helical scan VTR (Video Tape
Recorder). The first commercial PCM/digital recording
session was performed by DENON in 1972
(Takeaki, 1989).

During digital recording of the analog signal,
analog to digital (A/D) conversion takes place from
continuous time-amplitude coordinates to discrete
time-amplitude coordinates as illustrated in Figure 1.
The difference between the instantaneous analog
signal and de digital representation is digital error.

Figure 1: Use of an A/D (or D/A) converter to
convert a continuous function (time-amplitude) to a
discrete function (discrete time - discrete amplitude).
Conversion introduces a digital error in the signal -
digital noise.

We will separately consider the consequences
of discrete time and discrete amplitude coordinates
on the representation of the analog signal.

Discrete Time

Nyquist theorem

. The Nyquist theorem states
that if a signal V(t) does not contain frequencies higher
than fs/2 (where fs = 1/Ts),
then it can be fully recovered from its sampled values V( nTs)
at discrete times tn = nTs where n = ... -1, 0 , 1 , 2 , 3 ...

(1)

where:

fs = 1/Ts, the sampling frequency

V(t) = value of signal at arbitrary time t.

This is a remarkable result. The recovered
signal will have all the frequencies in the range from
0 to fs/2 Hz.

Discrete Amplitude

The term bit stands for binary digit and is associated
with a two-choice situation (0 and 1). Thus, any
digital system with just two levels has a 1 bit resolution.
Generally, the logarithm to the base 2 is used
to convert the number of available quantization
levels to number of bits. A device with two stable
positions, such as a relay or a flip-flop, can store 1
bit of information. N such devices can store N bits
of information, because the total number of possible
states is 2N and amount of information is equal to
log22N = N bits ( Shannon, 1949/1975). Thus, 4
levels is 2 bits, 8 is 3 bits, 16 is 4 bits, etc. For an
N-bit A/D or D/A converter

No. of levels = 2N

(2)

Example:

N = 8

No. of levels = 256

N = 12

No. of levels = 4,096

N = 16

No. of levels = 65,536

N = 20

No. of levels = 1,048,576

When a voltage amplitude from 0 to Vmax is used
(for example from 0 to 1 Volt), then one quantization step will be:

= Vmax / No. of levels =
Vmax/2N

(3)

At an adequately high level and complexity of input
signal V(t), the digital error (difference between
analog signal and stored digital value) from sample
to sample will be statistically independent and
uniformly distributed in the range of [ -/2, /2 ]
where is the step size in the A/D converter.

Thus, the maximum Signal-to-Noise Ratio (S/N) in
decibels can be calculated to be (Nakajima, 1983;
Mieszkowski, 1987):

A block diagram of a digital
recording/processing system is shown in figure 2.
The processes at each of the numbered blocks 1 to 7
are described below:

Figure 2: Block diagram of digital recordinglprocessing system. Both sources of noise [N1(t), N2(t)] are needed in
order to avoid digital distortions of the signal V(t) in the form of coherent noise ND(t). Properly chosen N1(t) and
N2(t) add only a little noise to the output, but remove coherence of ND(t) (digital noise) with the signal V(t).

Following Nakajima (1983), Mieszkowski (1989)
and Wannamaker, Lipshitz and Vanderkooy (1989),
analog dither must be added to the input signal in
order to

a) linearize the A/D converter

b) make possible improvement of S/N by averaging
process according to formula:

If DSP is performed on the signal, one must add digital dither N2(t) (box 5) to
avoid digital distortions and coherent noise ND (t) on the output of D/A converter.
Digital processing should also be performed using sufficiently precise real numbers to
avoid round-off errors.

Storage of digital data can be performed on
magnetic tape, optical disk, magnetic disk, or RAM
(Random Access Memory). Prior to storage, extra
code is generated to allow for error correction. This
error correction code allows detection and correction
of errors during playback of the audio signal.
Redundant information must be added to the original
signal in order to combat noise inherent in any
storage/communication system. The particular type
of code and error correction system depends on
storage medium, communication channel used and
immunity from errors (an arbitrarily small probability
of error can be obtained, Nakajima, 1983;
Shannon, 1949/1975).

Table I summarizes the author's comparison of
studio quality reel-to-reel analog tape recorder with 16
bit digital recorder. These data are derived from specifications
by various manufacturers of analog and
digital audio products. This table implies that the
digital recorder has many advantages over its analog
counterpart Performance of the analog recorder
depends very much on the calibration and tape used, as
well as on the environmental conditions such as temperature and humidity.
This is not the case for a digital
recorder, as long as errors generated are within the
limits of error correctability of the particular device.

Below is a short list of commonly used digital
coding algorithms (using as an example a single
channel digital recording system with swnpling
frequency fs = 44,100 Hz and 16 bit A/D and D/A
conversion). The data compression algorithms, which
are more efficient than PCM (use less storage
space), preserve the information content of the
signal. Not mentioned here are data reduction/compression
algorithms, which reduce information
content of the original signal (arbitrarily or on the
basis of psychoacoustics research results).

PCM
- PCM was invented by A.H. Reeves in
1939 (American Patents 2272070, 1942-2 see Nakajima, 1983)
and was analyzed and developed as a
modulation system from the point of view of communication theory by C.E.
Shannon (1949). Using
only two alternative pulse values (0 and 1), a 16-
pulse train is generated which indicates the sampled
value (for example, 1010 1111 0110 1101, a binary
coded 16 bit number). During conversion, 16 bit
amplitudes A1, A2, A3 ... are generated with a rate
44,100/sec. The demand on the storage device and
speed of transmission channel is 88,200 Bytes/sec.
This is a 'brute force' approach, which is not the
most effective way of using the storage device and
transmission channel.

ADPCM
- Adaptive Differential Pulse Code Modulation.
Depending on the signal, the number of available bits to represent
the difference between consecutive 16 bit samples is varied.
For example, for the
case of total quiet at the input (or small signal) the
difference could be switched off totally or represented
only by 1 bit. Demand on the storage device and the
speed of transmission channel could vary between 0
Bytes/sec and 88,200 Bytes/sec depending on signal
complexity. This is probably the most effective way of
coding. Similar means of coding could be used for
video signals because there is not much change from
frame to frame most of the time.

M
- Delta Modulation. During coding only 1 bit
differences between consecutive amplitudes are generated
at a high conversion speed indicating whether the
signal was increased or decreased (from the previous
sample). Demand on the storage device and the speed
of transmission channel is very high in comparison to
the PCM system for the same quality of signal (Nakajima, et al., 1983).

Recording/Storage Systems

Listed below are current common recording/storage
systems for digital audio data.

PCM unit + VCR recorder
- 2 and 4 channels.
These are professional and semi-professional systems
with 14 bit or 16 bit resolution and 44,056 Hz or
44,100 Hz sampling frequency. The PCM signal is
stored on video tape in pseudo-video format. Most of
the early systems were of this type.

DASH
(Digital Audio Stationary Head Recorder)
This is a professional 16 bit system with up to 48
tracks. Available are 40,056, 44,100 and 48,000 Hz
sampling frequencies.

R-DAT
(Rotating Head Digital Audio Tape Recoder)
This is a professional and consumer 2-channel system
with 16 bit resolution and 32,000, 44,056, 44,100,
48,000 Hz sampling frequencies.

Magnetic Hard Disk and RAM
(Random Access
Memory) based Recorders.
These are computer based
professional and semi-professional recording systems
having from 1 to 24 tracks. The resolution is from
8 to 18 bits. Sampling frequencies are from 2 kHz
to 250 kHz. The computers may be common
microcomputers as well as main-frame computers.
They offer the highest flexibility in terms of digital
editing of stored sound and in the author's opinion
are the trend of the future.

Optical WMRM (Write Many Read Many),
Erasable Optical Disk based Recorders.
This format is becoming popular for audio applications
because the removable optical cartridge can store
about 600 MBytes of data and is more robust than
magnetic media. Writing and reading is done by
laser without physical contact with the disk. The
NeXT computer has the first commercially available
optical disk drive with 256 MBytes capacity (
Thmpson and Baran, 1988). Also, Nakamichi recently
showed during the AES 7th International Conference
a working prototype of an optical disk recorder,
similar to a CD player (Mascenik, 1989).

Digital techniques for storing and transmission
of audio signals are attractive because they offer
high quality signals, which do not deteriorate with
transmission distance, number of copies or time.
Digital information when properly stored and transmitted
maintains its 100% integrity in contrast to
analog information which deteriorates during each
transmission and storage cycle.

DSP

is also far more powerful than ASP
(Analog Signal Processing). First, the quality of the
signal is maintained during DSP. Second, most of
the DSP devices are very flexible because one can
run many different applications on the same hardware
by a change of the software. Analog devices
are devoted to particular tasks and are not as flexible.
Third, digital signal processing can perform
operations impossible in the analogue domain.

Some of the functions which could be performed
by the DSP devices are: filtering, equalization,
compression/expansion of dynamic range, time
compression/expansion, delay, reverberation, pitch
change, generation of arbitrary signal or noise, music
and voice synthesis, noise reduction, signal restoration,
automatic pattern and voice recognition, time-
reverse, noise gate, automatic gain control, mixing
of signals, and FFT (Fast Fourier Transforms).

In recent years DSP units have become relatively
affordable. Also, there are many products available as
plug-in cards for popular microcomputers, which
contain DSP chips from such manufacturers as Motorola
or Texas Instruments. DSP systems based on
microcomputers are relatively fast (but not as fast as
devoted hardware) and very flexible.

The future of digital recording and DSP looks very
bright. Higher speeds of microprocessors and DSP
chips make real time applications of even complex
algorithms realistic. Falling prices of RAM chips and
storage devices like the erasable optical disk make
them affordable for many researchers and musicians.

In the author's opinion it is almost certain that the
majority of future recording and DSP equipment will
be based on microcomputers. Storage media of the
future will probably be erasable optical disks and RAM
cards. With falling prices of RAM chips and already
available 4 Mbit chips in a single package, one can
expect portable RAM based ADPCM recorders to
replace mechanically complex R-DAT machines in the
near future.